Modular Forms, Elliptic Curves, and their Connection to Fermat's Last Theorem Dylan Johnson ABSTRACT Fermat's Last Theorem (FLT) states that if n is an integer greater than three, the equation xn + yn = zn has no integer solutions with xyz 6= 0. This incredible statement eluded proof for over three-hundred years: in that time, mathematicians developed numerous tools which finally proved FLT in 1995. In this paper, we introduce some of the essential objects which enter the proof | especially modular forms, elliptic curves, and Galois representations | with an emphasis on precisely stating the Shimura-Taniyama Conjecture and explaining how its proof finally settled FLT. We offer proofs whenever they clarify a definition or elucidate an idea, but generally prefer examples and exposition which make concrete a truly beautiful body of mathematical theory. i Contents 0 Introduction 1 1 Some Algebraic Number Theory 3 1.1 Unique Factorisation and the Ring of Integers . .3 1.2 Inverse Limits and the Absolute Galois Group . .5 2 Modular Forms 8 2.1 The Group Action . .8 2.2 An Early Definition and Examples . 11 2.3 Congruence Subgroups . 12 3 Hecke Operators and Newforms 16 3.1 The Double Coset Operator . 16 3.2 The diamond Operator and Dirichlet Characters . 17 3.3 The Tp Operator . 21 3.4 Newforms . 22 4 Elliptic Curves 26 4.1 Projective Space . 27 4.2 Elliptic Curves . 28 4.3 Torsion and the Tate Module . 31 4.4 Reduction of Elliptic Curves over Q ....................... 33 5 Galois Representations 35 5.1 Frobenius Elements and Galois Representations . 35 5.2 Galois Representation Associated to an Elliptic Curve . 38 5.3 Galois Representation Associated to a Newform . 41 6 The Shimura-Taniyama Conjecture 43 7 Overview of the Proof of Fermat's Last Theorem 47 References 49 ii 0 Introduction Proven in 1995, Fermat's Last Theorem (FLT) remains a celebrity of twentieth century mathematics. FLT states that for n ≥ 3, the equation xn + yn = zn has no integer solutions with xyz 6= 0. The saga of FLT began in 1637 when Pierre de Fermat | a French lawyer who enjoyed mathematics in his spare time | conjectured his theorem in the margin of an old math book. Fermat wrote that he had found a \truly marvelous proof" of the theorem, but that the book's margin was simply \to narrow to contain it". Because Fermat never published a formal proof | and it took mathematicians over three hundred years to devise one | it seems almost certain that Fermat never actually proved his own theorem. Nevertheless, he sparked a wildfire: countless mathematicians developed incredible mathematics in an effort to prove FLT. Early attempts relied on a crucial simplifying observation: a non-trivial solution | i.e. a solution with xyz 6= 0 | to the equation xpd + ypd = zpd for the exponent pd yields a non-trivial solution (xd)p + (yd)p = (zd)p for the exponent p. Because Fermat did indeed prove his theorem in the case of n = 4, the observation shows that proving FLT for any exponent n ≥ 3 reduces to proving FLT for equations xp + yp = zp with p an odd prime. So the first progress on FLT involved checking the statement for p = 3 (by Euler in 1770), p = 5 (by Legendre and Dirichlet, independently, around 1825), and p = 7 (by Lam´ein 1865). The first substantial case of FLT came from the work of Sophie Germain in 1823. Germain introduced a much more general strategy for attacking the problem which ultimately split the problem into two cases: 1. Case 1 is the non-existence of xp + yp = zp for which p doesn't divide xyz; and 2. Case 2 is the same but when p does divide xyz. Together with Legendre, Germain applied her techniques to prove the first case of FLT for all primes less than or equal to 197 (see [Rid09] for additional details). In particular, Germain showed that the first case of FLT holds for odd \Germain primes": an odd prime p such that 2p + 1 is also prime. It remains unknown whether or not there exist infinitely-many Germain primes, so we still aren't sure if Germain's results gave an infinitude of results on FLT. The next substantial case of FLT came from Ernst Kummer who drew upon ideas by Lam´eon unique factorisation. Kummer proved (both case 1 and case 2) of FLT for so-called \regular primes" around 1850. The precise definition of a regular prime relies on the notion of factorisation of ring ideals; in section 1, we introduce some of these ideas (and defer to [Mil17] for the rest). Although certain heuristics predict that roughly 61% of primes are regular, whether or not there are infinitely-many remains a mystery. So two hundred years after Fermat conjectured his theorem, mathematicians still weren't sure about infinitely many cases. The twentieth century witnessed the development of a wonderful body of mathematics which would go on to prove Fermat's Last Theorem. Of central importance are the ideas of Gor¯oShimura and Yutaka Taniyama | who conjectured a precise relationship between \modular forms" and \elliptic curves" | and Jean-Pierre Serre | who conjectured a precise relationship between \modular forms" and \Galois representations". Ultimately, the link 1 between modular forms and elliptic curves became an invaluable tool and by 1990 it was known that Fermat's Last Theorem would follow from the Shimura-Taniyama Conjecture. Andrew Wiles thus proved FLT by proving (most of) Shimura-Taniyama. In this paper, we offer a broad overview of the twentieth century mathematics which proved FLT; we emphasise the role of the Shimura-Taniyama Conjecture (STC) in the proof and indeed develop the necessary language to precisely state STC. Along the way, we intro- duce some of number theory's most important tools and techniques. While we provide many proofs (especially in sections 2 and 5), we generally prefer illustrative examples to technical arguments. Section 1 introduces some essential algebraic number theory (material from [Mil17] and [Mil18]), and is intended mostly as a reference for later sections. Section 2 introduces modular forms and their relationship with subgroups of SL2(Z) (material from chapters 1 and 2 of [DS05]). Section 3 introduces Hecke operators as well as newforms, the modular forms which play a role in STC (material from chapter 5 of [DS05]). Section 4 introduces elliptic curves, including the Tate module of an elliptic curve and reduction of curves over Q (material from chapters 3 and 7 of [Sil86]). Section 5 shifts to the construction of Galois representations for both newforms and ellip- tic curves, as well as discusses the structure of the absolute Galois group Gal(Q=Q) (material from chapter 8 of [Mil17] and chapter 9 of [DS05]). Finally, section 6 states the Shimura- Taniyama Conjecture and goes through an explicit example to illustrate the statement (ma- terial from chapter 9 of [DS05] and [Wes99]). Section 7 then brings everything together by outlining the stepping stones which go into proving FLT (material from [DDT07]). 4. Elliptic Curves 1. Some Algebraic 5. Galois Rep- Number Theory resentations 3. Hecke Operators 2. Modular Forms and Newforms 6. The Shimura- Taniyama Conjecture 7. Overview of the Proof of Fermat's Last Theorem 2 1 Some Algebraic Number Theory Developed throughout the nineteenth and early twentieth century, basic algebraic number theory forms the foundation for not only Lam´eand Kummer's early attempts at FLT, but also for the Shimura-Taniyama Conjecture and Wiles's ultimate proof. In this section, we highlight the essential results we will need later, and intend that this section serves only as a reference for subsequent sections. 1.1 Unique Factorisation and the Ring of Integers One of the greatest structural features of the integers is that they permit unique factori- sation. Indeed, a common strategy for attacking problems over Z is to first consider the problem for irreducible (prime) integers | which are often easier to understand | before then assembling information for a general integer from information on its prime divisors. But unique factorisationp fails inp rings \notp too much larger" than the integers: for example, 6 = 2 · 3 = (1 + −5)(1 − −5) in Z[ −5]. Roughly speaking, the algebraic number theory of the late 1800s worked to recover a notion of unique factorisation in more general rings. Ultimately, the solution is to consider factorisation of ring ideals, rather than of ring elements. We begin with some essential algebraic notions. Throughout this section, we will work over Q | as Q is our principal concern as number theorists | keeping in mind that many of these ideas naturally generalise to arbitrary rings/fields. Definition 1.1. A number α 2 C is an algebraic number if there exists a polynomial n n−1 f(x) = x + an−1x + ··· + a1x + a0; ai 2 Q such that f(α) = 0. We denote by Q the set of all algebraic numbers, where the notation comes from the algebraic closure of a field; that is, Q may equivalently be thought of as the algebraic closure of Q, just as C is the algebraic closure of R. Similarly, we define an algebraic integer by replacing Q with Z: Definition 1.2. A number α 2 C is an algebraic integer if there exists a polynomial n n−1 f(x) = x + an−1x + ··· + a1x + a0; ai 2 Z such that f(α) = 0. We denote by Z the set of all algebraic integers, where the notation comes from a more general notion of \integral closure". Finally, rather than consider all algebraic numbers Q, we will want to consider finite subfields.